The Problem of Time in Quantum Cosmology
and Non-chronometric Temporality
by
Carlos Pedro Gonçalves
Mathematics researcher at UNIDE-ISCTE
cpdsg.research@gmail.com (primary); cpdsg@iscte.pt
Maria Odete Madeira
Interdisciplinary researcher in philosophy of science and systems science
mosmg.research@gmail.com (primary), mariaodete.sm@gmail.com
Abstract
"We review two lines of argument regarding the problem of time in quantum cosmology and in
quantum gravity, one that invokes the path integral formalism for quantum gravity to state the
absence of time between two three-geometries, and another that defends the absence of time, as
a fundamental notion in physics, in terms of: (a) the configuration space argument , put forward
by Barbour, Smolin and Kauffman, and (b) the Wheeler-DeWitt equation.
quantum gravity, one that invokes the path integral formalism for quantum gravity to state the
absence of time between two three-geometries, and another that defends the absence of time, as
a fundamental notion in physics, in terms of: (a) the configuration space argument , put forward
by Barbour, Smolin and Kauffman, and (b) the Wheeler-DeWitt equation.
We argue that although being correct with respect to a space-time dependent physical
chronometrizable clock-time frame, both of these lines of argument fail with respect to a general
sense of temporality, expressed in terms of the more elementary notions of a before and an after
of a quantum computation.
With respect to the first line of argument, it is shown that the early works on the subject
address two kinds of temporalities, one that is the space-time geometric dependent temporality,
which coincides with the usual definition of a space-time dependent physical chronometrizable
clock-time frame, the other is a temporality associated to the notions of input and output of a
general quantum gravity computation, that is expressed, in the theoretical discourse of quantum
gravity, through the usage of the concepts of: (1) propagation of a wave functional in superspace,
as addressed by Wheeler; (2) transition amplitudes of three-geometries and (3) the pathintegral
formalism, used to calculate such amplitudes, as addressed by Hartle and Hawking.
While the first temporality (space-time dependent temporality) disappears from the theory, the
second plays a fundamental role, not only in the several aspects of the theory’s construction, but
in the clock-time independence as well, as Wheeler showed.
address two kinds of temporalities, one that is the space-time geometric dependent temporality,
which coincides with the usual definition of a space-time dependent physical chronometrizable
clock-time frame, the other is a temporality associated to the notions of input and output of a
general quantum gravity computation, that is expressed, in the theoretical discourse of quantum
gravity, through the usage of the concepts of: (1) propagation of a wave functional in superspace,
as addressed by Wheeler; (2) transition amplitudes of three-geometries and (3) the pathintegral
formalism, used to calculate such amplitudes, as addressed by Hartle and Hawking.
While the first temporality (space-time dependent temporality) disappears from the theory, the
second plays a fundamental role, not only in the several aspects of the theory’s construction, but
in the clock-time independence as well, as Wheeler showed.
Given this notion of time, different from a chronometrizable, space-time geometry internal
notion, we search for a general mathematical and logical structure that is capable of addressing
it from a formal point of view. This is done through a family of mathematical structures that is
more general than the mathematical category. These structures not only will allow us to address
the nature of the temporality present in the transition amplitudes between two three-geometries, but they will also allow us to refute the configuration space argument and to show how a static clock-time-independent quantum state, can be put into a non-clock-time processual expression in terms of fine-grained computational histories, obtained from the relations between different observable’s bases."
notion, we search for a general mathematical and logical structure that is capable of addressing
it from a formal point of view. This is done through a family of mathematical structures that is
more general than the mathematical category. These structures not only will allow us to address
the nature of the temporality present in the transition amplitudes between two three-geometries, but they will also allow us to refute the configuration space argument and to show how a static clock-time-independent quantum state, can be put into a non-clock-time processual expression in terms of fine-grained computational histories, obtained from the relations between different observable’s bases."
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